David L Miller (@millerdl)
University of Melbourne
15 April 2016
Slides available at http://converged.yt
(a talk)
\[ \mathbb{P} \left[ \text{animal detected } \vert \text{ animal at distance } y\right] = g(y;\boldsymbol{\theta}) \]
\[ \hat{p} = \frac{1}{w} \int_0^w g(y; \boldsymbol{\hat{\theta}}) \text{d}y \]
Figure from Marques et al (2007)
\[ \hat{N} = \frac{\text{study area}}{\text{survey area}}\sum_{i=1}^n \frac{s_i}{\hat{p}_i} \]
(where \( s_i \) are group/cluster sizes)
Rather than hoping our design detects to the changes in density and contriving a complex set of strata, use a model!
Hedley and Buckland (2004)
Miller et al. (2013)
Taking the previous example…
\[ n_j = \color{red}{A_j}\color{blue}{\hat{p}_j} \color{green}{\exp}\left[\color{grey}{ \beta_0 + \sum_k f_k(z_{kj})} \right] + \epsilon_j \]
where \( \epsilon_j \sim N(0, \sigma^2) \), \( \quad n_j\sim \) count distribution
library(dsm)
dsm_env_tw <- dsm(count~s(Depth) + s(NPP) + s(SST),
ddf.obj=df_hr,
segment.data=segs, observation.data=obs,
family=tw(), method="REML")
(method="REML" uses REML to select wigglyness; Wood, 2011)
dsm is based on mgcv by Simon Wood
“Everything is related to everything else, but near things are more related than distant things”
Tobler (1970)
mgcv::concurvity)
\[ n_j = A_j p_j \exp \left\{ \color{red}{\left[\frac{ \partial \log p_j(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}} \Big\vert_{\boldsymbol{\theta} = \hat{\boldsymbol{\theta}}}\right] \boldsymbol{\gamma}} + \beta_0 + \sum_k f_k(z_{jk}) \right\} \]
Picture from University of St Andrews Library Special Collections
Slides w/ references available at converged.yt